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INEXACT NEWTON METHODS
A classical algorithm for solving the system of nonlinear equations $F(x) = 0$ is Newton’s method \[ x_{k + 1} = x_k + s_k ,\quad {\text{where }}F'(x_k )s_k = - F(x_k ),\quad x_0 {\text{ given}}.\]...
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Variational Iterative Methods for Nonsymmetric Systems of Linear Equations
TLDR
A class of iterative algorithms for solving systems of linear equations where the coefficient matrix is nonsymmetric with positive-definite symmetric part, modelled after the conjugate gradient method, are considered.
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A Supernodal Approach to Sparse Partial Pivoting
TLDR
A sparse LU code is developed that is significantly faster than earlier partial pivoting codes and compared with UMFPACK, which uses a multifrontal approach; the code is very competitive in time and storage requirements, especially for large problems.
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Choosing the Forcing Terms in an Inexact Newton Method
TLDR
Promising choices of the forcing terms are given, their local convergence properties are analyzed, and their practical performance is shown on a representative set of test problems.
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Efficient Algorithms for Computing a Strong Rank-Revealing QR Factorization
TLDR
Two algorithms are presented for computing rank-revealing QR factorizations that are nearly as efficient as QR with column pivoting for most problems and take O (ran2) floating-point operations in the worst case.
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Gmres: a Generalized Minimum Residual Algorithm for Solving
Overton, who showed us how the ideal Arnoldi and GMRES problems relate to more general problems of minimization of singular values of functions of matrices 17]. gmres and arnoldi as matrix…
Globally Convergent Inexact Newton Methods
TLDR
The primary goal is to introduce and analyze new inexact Newton methods, but consideration is also given to “globally convergence” features designed to improve convergence from arbitrary starting points.
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Computing the Singular Value Decomposition with High Relative Accuracy
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A Divide-and-Conquer Algorithm for the Symmetric Tridiagonal Eigenproblem
TLDR
A new, stable method for finding the spectral decomposition of a symmetric arrowhead matrix and a new implementation of deflation are presented, which are competitive with bisection with inverse iteration, Cuppen's divide-and-conquer algorithm, and the QR algorithm for solving the symmetric tridiagonal eigenproblem.
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Efficient Implementation of a Class of Preconditioned Conjugate Gradient Methods
The preconditioned conjugate gradient (PCG) method is an effective means for solving systems of linear equations where the coefficient matrix is symmetric and positive definite. The incomplete $LDL^t…
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